Using the conservation laws for charge, energy, momentum, and angular momentum, we derive hydrodynamic equations for the charge density, local temperature, and fluid velocity, as well as for the polarization tensor, starting from local equilibrium distribution functions for particles and antiparticles with spin $1/2$. The resulting set of differential equations extends the standard picture of perfect-fluid hydrodynamics with a conserved entropy current in a minimal way. This framework can be used in space-time analyses of the evolution of spin and polarization in various physical systems including high-energy nuclear collisions. We demonstrate that a stationary vortex, which exhibits vorticity-spin alignment, corresponds to a special solution of the spin-hydrodynamical equations.

Relativistic hydrodynamics has been quite successful in explaining the collective behaviour of the QCD matter produced in high energy heavy-ion collisions at RHIC and LHC. We briefly review the latest developments in the hydrodynamical modeling of relativistic heavy-ion collisions. Essential ingredients of the model such as the hydrodynamic evolution equations, dissipation, initial conditions, equation of state, and freeze-out process are reviewed. We discuss observable quantities such as particle spectra and anisotropic flow and effect of viscosity on these observables. Recent developments such as event-by-event fluctuations, flow in small systems (proton-proton and proton-nucleus collisions), flow in ultracentral collisions, longitudinal fluctuations, and correlations and flow in intense magnetic field are also discussed.

Starting from the Boltzmann equation with the relaxation time approximation for the collision term and using a Chapman-Enskog-like expansion for the distribution function close to equilibrium, we derive hydrodynamic evolution equations for the dissipative quantities directly from their definition. Although the form of the equations is identical to those obtained in traditional Israel-Stewart approaches employing Grad's 14-moment approximation and the second moment of the Boltzmann equation, the coefficients obtained are different. In the case of a one-dimensional scaling expansion, we demonstrate that our results are in better agreement with a numerical solution of the Boltzmann equation as compared to Israel-Stewart results. We also show that including approximate higher-order corrections in viscous evolution significantly improves this agreement, thus justifying the relaxation time approximation for the collision term.

We present the derivation of a novel third-order hydrodynamic evolution equation for the shear stress tensor from kinetic theory. The Boltzmann equation with a relaxation time approximation for the collision term is solved iteratively using a Chapman-Enskog-like expansion to obtain the nonequilibrium phase-space distribution function. Subsequently, the evolution equation for the shear stress tensor is derived from its kinetic definition up to third order in gradients. We quantify the significance of the new derivation within a one-dimensional scaling expansion and demonstrate that the results obtained using the third-order viscous equations derived here provides a very good approximation to the exact solution of the Boltzmann equation in a relaxation time approximation. We also show that the time evolution of pressure anisotropy obtained using our equations is in better agreement with transport results than that obtained with an existing third-order calculation based on the second law of thermodynamics.

Recently advocated expressions for the phase-space dependent spin-$\frac{1}{2}$ density matrices of particles and antiparticles are analyzed in detail and reduced to forms linear in the Dirac spin operator. This allows for a natural determination of the spin-polarization vectors of particles and antiparticles by evaluating the trace of products of the spin density matrices and the Pauli matrices. We demonstrate that the total spin-polarization vector obtained in this way agrees with the Pauli-Lubański four-vector, constructed from an appropriately chosen spin tensor and boosted to the particle rest frame. We further show that several forms of the spin tensor used in the literature yield the same Pauli-Lubański four-vector.